If we assume that a set of operationally defined statistics is produced by a quantum causal structure taking this specific form, we can then use statistical tools to make inferences about the nature of the underlying quantum structure - for example, given a process operator we can use the algorithm set out in ref [1]} to find out if the operator obeys the Markov condition for some particular DAG on the quantum nodes, thus determining whether that DAG represents a candidate quantum causal structure for the process. Evidently there are a number of practical uses for this formalism - for example, given an experimental procedure where we are not sure our apparatus is doing exactly what we expected, we can apply quantum causal inference to determine if the observed statistics are consistent with our desired underlying quantum structure. However, it has also been suggested that there may be foundational applications of this formalism - it serves to `establish an account of causality in quantum theory’s own terms, without assuming a separate realm of classical systems or measurement outcomes. '[1]} One of our aims in this article will be to evaluate these claims.
